A fire breaks out and increases the Kelvin temperature of a cylinder of compressed gas by a factor of 1.4. What is the final pressure of the gas relative to its initial pressure?
since PV = kT,
if T grows by 1.4, so does P, if V doesn't change
Well, it sounds like this gas cylinder is really heating things up, quite literally! Now, let's calculate the final pressure of the gas relative to its initial pressure.
To do that, we can use the ideal gas law, which states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.
Since the volume and number of moles of the gas remain constant, we can simplify the equation to P1/T1 = P2/T2, where P1 is the initial pressure, T1 is the initial temperature, P2 is the final pressure, and T2 is the final temperature.
Now, you mentioned that the Kelvin temperature of the gas increased by a factor of 1.4. This means that T2 = 1.4 * T1.
Plugging that into the equation, we have P1/T1 = P2 / (1.4 * T1).
Buckle up because things are about to get a bit mathematical! To get P2 by itself, we can cross multiply and rearrange the equation to solve for P2.
This gives us: P2 = (P1 * 1.4 * T1) / T1.
Simplifying further, we find that P2 = 1.4 * P1.
So, the final pressure of the gas relative to its initial pressure is 1.4 times the initial pressure. It's like the pressure is saying, "I'm feeling 1.4 times hotter now!"
To determine the final pressure of the gas relative to its initial pressure, we can use the Gay-Lussac's Law of combining volumes of gases.
According to Gay-Lussac's Law, the pressure of a gas is directly proportional to its temperature when the volume and amount of gas remain constant.
Mathematically, the relationship can be expressed as:
(P1 / T1) = (P2 / T2)
Where:
P1 = initial pressure
T1 = initial temperature
P2 = final pressure
T2 = final temperature
Given that the Kelvin temperature of the gas increases by a factor of 1.4, we can write:
T2 = 1.4 * T1
Substituting this value into the Gay-Lussac's Law equation, we have:
(P1 / T1) = (P2 / (1.4 * T1))
Simplifying the equation, we find:
P2 = 1.4 * P1
Therefore, the final pressure of the gas relative to its initial pressure is 1.4 times its initial pressure.
To find the final pressure of the gas, we need to understand the relationship between temperature and pressure according to the ideal gas law.
The ideal gas law states: PV = nRT
Where:
P is the pressure of the gas
V is the volume of the gas
n is the number of moles of the gas
R is the ideal gas constant
T is the temperature of the gas in Kelvin
Since we are given the ratio of the temperatures, we can assume that the volume, number of moles, and volume remain constant. Therefore, we can rewrite the equation as a ratio:
P1V1 / T1 = P2V2 / T2
where:
P1 is the initial pressure of the gas
T1 is the initial temperature of the gas in Kelvin
P2 is the final pressure of the gas
T2 is the final temperature of the gas in Kelvin
Since the volume and number of moles remain constant, we can simplify the equation:
P1 / T1 = P2 / T2
Now, we can plug in the given information:
T2 / T1 = 1.4
P1 / T1 = P2 / T2
P1 / T1 = P2 / (1.4 * T1)
P1 = P2 / 1.4
Therefore, the final pressure of the gas relative to its initial pressure is 1/1.4, or approximately 0.714 times the initial pressure.